GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 20 Jun 2018, 20:02

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Expert Post
8 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 46207
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 23 Aug 2015, 22:49
8
34
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

59% (01:30) correct 41% (01:15) wrong based on 517 sessions

HideShow timer Statistics

3 KUDOS received
Intern
Intern
avatar
Joined: 28 Jan 2013
Posts: 31
Location: United States
Concentration: General Management, International Business
GPA: 3.1
WE: Information Technology (Consulting)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 23 Aug 2015, 23:09
3
6
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

1/2 * (2^30 + 2^30) = 2^30

2^29 + 2^29 = 2^30

The above equation is similar to 2^x + 2^y =2^30.
so x =29 and y=29

Ans:D
6 KUDOS received
Senior Manager
Senior Manager
User avatar
B
Joined: 21 Jan 2015
Posts: 286
Location: India
Concentration: Strategy, Marketing
GMAT 1: 620 Q48 V28
WE: Sales (Consumer Products)
GMAT ToolKit User CAT Tests
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 23 Aug 2015, 23:22
6
5
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64


Ans: D

Solution: 2^x + 2^y =2^30
there must be some value of as we know 2^30 only has 2 as its factor so if i take anything common from LHS of the equation the remaining part must be in form of 2 only.
I can take common if and only if x=y; otherwise the remaining part inside the bracket will become odd
for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y
so now=>
the equation becomes 2^x + 2^x =2^30 because x=y
2^x (1+1)= 2^30
2^x * 2 = 2^30
x+1 = 30
x= 29
and y = 29
x+y = 58
_________________

--------------------------------------------------------------------
The Mind is Everything, What we Think we Become.
Kudos will encourage many others, like me.
Please Give Kudos Image !!
Thanks :-)

5 KUDOS received
Manager
Manager
User avatar
Joined: 14 Mar 2014
Posts: 148
GMAT 1: 710 Q50 V34
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 24 Aug 2015, 00:48
5
4
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.


IMO: D

Both sides consists of only powers of 2.

Thus in order to that happen x = y must be the case

if x=y then

\(2^x +2^y =2 ^30\) ==> \(2^x +2^x =2 ^30\)
\(2(2^x) =2 ^30\)
x+1 =30
x=29
Thus x+y = 29+29 =58
_________________

I'm happy, if I make math for you slightly clearer
And yes, I like kudos
¯\_(ツ)_/¯ :-)

8 KUDOS received
Manager
Manager
avatar
Joined: 15 May 2014
Posts: 64
GMAT ToolKit User Premium Member
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 24 Aug 2015, 21:38
8
7
\(2^x+2^y\,=\,2^{30}\)
\(x+y\,=\,?\)

\(2^x\,(1+2^{y-x})\,=\,2^{30}\)
\(1+2^{y-x}\,=\,2^{30-x}\)
\(1\,=\,2^{30-x}-2^{y-x}\)

Difference between any two powers of 2 yield 1 if one of the powers is one and the other is zero
i.e. \(2^1\,-\,2^0\,=\,1\)

--> \(30-x\,=\,1\,\,\,\) and \(\,\,\,y-x\,=\,0\)
\(x\,=\,29\,\,\,\) and \(\,\,\,x\,=\,y\)
\(x+y\,=\,58\)

Answer D
5 KUDOS received
Manager
Manager
avatar
Joined: 15 May 2014
Posts: 64
GMAT ToolKit User Premium Member
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 25 Aug 2015, 01:43
5
2
2^1 = 2^0 + 2^0
2^2 = 2^1 + 2^1
2^3 = 2^2 + 2^2
.
.
.
2^30 = 2^29 + 2^29

x + y = 58

Option D
1 KUDOS received
Director
Director
avatar
G
Joined: 21 May 2013
Posts: 646
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 25 Aug 2015, 12:15
1
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.


2^x+2^y=2^30
Now, 2^30=2^29+2^29=2*2^29
or x+y=29+29=58
Answer D
Intern
Intern
User avatar
Joined: 02 Jun 2015
Posts: 33
Location: United States
Concentration: Operations, Technology
GMAT Date: 08-22-2015
GPA: 3.92
WE: Science (Other)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 25 Aug 2015, 14:11
dkumar2012 wrote:
for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y


This is a great solution. A little more detail for those wanting to see why all cases will not work (not just the one above)

2^x + 2^y = 2^x * (1+2^(y-x)) = 2^30

This means that 1+2^(y-x) has to be a multiple of 2 and this can only be achieved if 2^(y-x) = 1 = 2^0.
Expert Post
2 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 46207
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 30 Aug 2015, 08:41
2
5
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.


VERITAS PREP OFFICIAL SOLUTION:

This problem is a good candidate for testing small numbers to find patterns. While your instincts may tell you to simply set x + y equal to 30, small numbers will show that you cannot simply set them equal. If you try \(2^x+2^y=2^6\), for example you can't find values for x + y = 6 that will set the sum equal to 64. The powers of 2 are:

2

4

8

16

32

64

The only pairing that will work is 32 + 32 = 64, meaning that you need \(2^5+2^5=2^6\), which should make sense: 2 times 2^5 will equal 2^6. So this example should teach you that you need to add two \(2^{29}\)s together to get to \(2^{30}\). So x and y are each 29, making the sum x + y equal to 58, answer choice D.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern
Intern
avatar
Joined: 03 Dec 2014
Posts: 15
GMAT 1: 620 Q43 V32
GPA: 2.9
GMAT ToolKit User Premium Member Reviews Badge
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 19 Nov 2015, 17:58
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.


2^x+2^Y = 2^30

2^x-30 + 2^y-30 = 1 (Divide LHS and RHS by 2^30)

1/2+1/2 = 1 Therefore 2^x-30=1/2 i.e. x-30 = -1 hence X= 29
similarly y-30 = -1 and y = 29

X+Y = 29+29 = 58

Ans D
1 KUDOS received
Current Student
avatar
S
Joined: 20 Mar 2014
Posts: 2643
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
GMAT ToolKit User Premium Member Reviews Badge
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 08 Jan 2016, 10:08
1
shasadou wrote:
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64


You are given 2^x+2^y=2^30 ---> in order to understand the question, try to see what values can y take if you start with fixed values for x ---> remember that x and y MUST be integers.

Thus, lets have x =1 ---> \(2^y=2^{30}-2\) ---> y can not be an integer. Try a bigger value for x.

x = 15 ---> \(2^y=2^{30}-2^{15}\) ---> \(2^y=2^{15}(2^{15}-1)\) , keep trying and you will see that there will always be a 1 inside the bracket except when you write the given expression as

\(2^x+2^y=2^{30}\) --->\(2^x+2^y=2*2^{29}\) ---> \(2^x+2^y=2^{29}+2^{29}\) ---> compare both sides of the equation to get x=y=29 and x+y = 58.

D is thus the correct answer.

Hope this helps.
Expert Post
6 KUDOS received
Veritas Prep GMAT Instructor
User avatar
P
Joined: 16 Oct 2010
Posts: 8102
Location: Pune, India
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 08 Jan 2016, 23:07
6
6
shasadou wrote:
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64


This is a property of exponents:

\(2^1 + 2^1 = 2^2\)
\(2^2 + 2^2 = 2^3\) (because \(2^2 * (1 + 1) = 2^2 * 2\))
Similarly, \(2^3 + 2^3 = 2^4\)
\(2^4 + 2^4 = 2^5\)
etc

So
\(2^{29} + 2^{29} = 2^{30}\)

\(x + y = 29 + 29 = 58\)

Similarly,

\(3^1 + 3^1 + 3^1 = 3^2\)
\(3^2 + 3^2 + 3^2 = 3^3\)
\(3^3 + 3^3 + 3^3 = 3^4\)
and so on...
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199

Veritas Prep Reviews

Expert Post
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 5899
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 11 Jan 2016, 09:12
shasadou wrote:
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64


Hi,
here we are adding two different numbers which are power to base 2 resulting into another number which has a base of 2..
the logic is .. if we are adding 2^x and 2^y to get 2^z, only way is to get 2 out of the addition and that is possible only when x and y are same..
so 2^x+2^y=2^30..where x =y
can be written as 2^x+2^x=2^30
2*2^x=2^30..
x+1=30 ..x=29..
so x+y=29+29=58
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html


GMAT online Tutor

1 KUDOS received
Intern
Intern
avatar
B
Status: Crushing with a 780
Joined: 26 Aug 2014
Posts: 19
GMAT ToolKit User Premium Member Reviews Badge
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 16 Apr 2016, 19:28
1
Here's a better way to solve the problem (suggested by Anthony Ritz 8-) - big thank you! ).

Step 1: factor out the given equation and see if there is a pattern
\(2^x(1 + 2^{y-x}) = 2^{30}\)
Hm, interesting. We know that \(2^x\) is always even, so \((1 + 2^{y-x})\) should also be even since \(2^{30}\) is even.

For \((1 + 2^{y-x})\) to be an even integer, \(2^{y-x}\) needs to be odd. What situation will \(2^{y-x}\) be an odd number? When \(2^{y-x} = 1\)!
Hence,\(y-x = 0\) and consequently \(x=y\).

Step 2: plug in our findings to the equation
\(2^x + 2^y = 2^x + 2^x = 2 * 2^x = 2^{1+x} = 2^{30}\)
The last two parts of our equation tells us that \(1+x = 30\).
In conclusion, \(x = y = 29\).

Answer: \(x + y = 58\).
Manager
Manager
avatar
Joined: 10 Apr 2016
Posts: 55
Concentration: Strategy, Entrepreneurship
GMAT 1: 520 Q29 V30
GPA: 3.01
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 17 Apr 2016, 03:42
It's actually really easy to spot a pattern here.

2^2 + 2^2 = 8
2^3 = 8
2^4 + 2^4= 16 + 16 = 32
2^5 = 32

So 2^x +2^y = 2^n
to get x+y you just do 2(n-1).
_________________

Took the Gmat and got a 520 after studying for 3 weeks with a fulltime job. Now taking it again, but with 6 weeks of prep time and a part time job. Studying every day is key, try to do at least 5 exercises a day.

Current Student
User avatar
B
Status: DONE!
Joined: 05 Sep 2016
Posts: 398
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 05 Nov 2016, 18:35
I used a simple tester example to see what would happen if I wanted to find a smaller exponential value of 2.

I used the following:

2^4 = 2^x + 2^y --> 8+8 (i.e. 2^3 + 2^3) satisfies this...

sum of x and y has to be one less than exponent we are looking for. Thus 29+29 = 58 and we have our answer.
Manager
Manager
User avatar
B
Joined: 19 Oct 2016
Posts: 73
Location: India
Concentration: Marketing, Leadership
Schools: IIMA (I)
GMAT 1: 580 Q46 V24
GMAT 2: 540 Q39 V25
GMAT 3: 660 Q48 V34
GPA: 3.15
WE: Psychology and Counseling (Health Care)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 05 Nov 2016, 23:34
Bunuel wrote:
For integers x and y, \(2^x + 2^y=2^{30}\). What is the value of \(x + y\)?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.


Love this as a 700 level question. Easy if you keep your head on straight. Plug in 2^2 + 2^2= 8 = 2^3.... Plug in 2^3 +2^3= 16 = 2^4....As we can see the pattern shows that 2^29 + 2 ^29 will equal 2^30. Then we can add 29 + 29 and get 58.

Under 30 seconds oooooooooooooooooooooh yea
Intern
Intern
avatar
B
Joined: 11 Dec 2016
Posts: 33
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 11 May 2018, 05:32
VeritasPrepKarishma wrote:
shasadou wrote:
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64


This is a property of exponents:

\(2^1 + 2^1 = 2^2\)
\(2^2 + 2^2 = 2^3\) (because \(2^2 * (1 + 1) = 2^2 * 2\))
Similarly, \(2^3 + 2^3 = 2^4\)
\(2^4 + 2^4 = 2^5\)
etc

So
\(2^{29} + 2^{29} = 2^{30}\)

\(x + y = 29 + 29 = 58\)

Similarly,

\(3^1 + 3^1 + 3^1 = 3^2\)
\(3^2 + 3^2 + 3^2 = 3^3\)
\(3^3 + 3^3 + 3^3 = 3^4\)
and so on...


Thanks for making things much easier!! Love your blogs as well. Keep up the great work.
Manager
Manager
User avatar
S
Joined: 29 Dec 2017
Posts: 218
Location: United States
Concentration: Marketing, Technology
GMAT 1: 630 Q44 V33
GPA: 3.25
WE: Marketing (Telecommunications)
CAT Tests
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 11 May 2018, 13:42
Alternative approach if you don's see the pattern:

\(2^x + 2^y=2^{30}\), where \(2^x + 2^y\)=\((2^x +2^y)^2-2*2^{x+y}=2^{30}\), where \((2^x + 2^y)^2=2^{60}\)

\(2^{30}*(2^{30}-1)=2^{x+y+1}\) ,

we can omit -1 since it almost doesn't change the result, so \(2^{60}= 2^{x+y+1}\), x+y+1=60. x+y = 59 - yes I know that this is not 58, but this is the closest result. Answer (D)
_________________

I'm looking for a study buddy in NY, who is aiming at 700+. PM me.

Manager
Manager
User avatar
S
Joined: 14 Dec 2017
Posts: 167
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y? [#permalink]

Show Tags

New post 08 Jun 2018, 02:06
Bunuel wrote:
For integers x and y, \(2^x + 2^y=2^{30}\). What is the value of \(x + y\)?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.



\(2^x + 2^y=2^{30}\)
which can be written as, \(2^{29}*({2^{x-29}} + {2^{y-29}})=2^{30}\)

Hence, \({2^{x-29}} + {2^{y-29}} = 2\)

Since x & y are integers, we got \(x-29 = 0\) & \(y-29 = 0\)

\(x = 29\), \(y =29\), \(x+y = 58\)

Answer D.


Thanks,
GyM
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?   [#permalink] 08 Jun 2018, 02:06

Go to page    1   2    Next  [ 21 posts ] 

Display posts from previous: Sort by

For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.